# Algebraic properties of trees (Acta Universitatis Carolinae - download pdf or read online

By Ladislav NebeskyÌ

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**Extra info for Algebraic properties of trees (Acta Universitatis Carolinae : Philologica monographia)**

**Example text**

It is thus intuitively very "small", but it is not a small set in our sense; {U} E U would imply U E U, a contradiction to the axiom of regularity. Thus "small set" for us means a member of the universe, and not a set with a small cardinal number. Our foundation by means of one universe does provide, within set theory, an accurate way of discussing the category of all small sets and all small groups, but it does not provide sets to represent certain metacategories, such as the metacategory of all sets or that of all groups.

S), also written (T, S), has as objects all triples

0 For two such arrows g : a ---+ hom (a', b) k* 1 hom (a', b') a' and k : b ---+ b' the diagram -"""g*-...... hom (a, 1 b) (6) k* g* • hom(a, b') in Set is commutative, because both paths send J E hom( a', b) to k J g. 35 Contravariance and Opposites These hom-functors have been defined only for a category C with small hom-sets. The familiar large categories Grp, Set, Top, etc. do have this property. To include categories without this property, we can proceed as follows: Given a category C, take a set V large enough to include all subsets of the set of arrows of C (for example, V could be the power set of the set of arrows of C).

### Algebraic properties of trees (Acta Universitatis Carolinae : Philologica monographia) by Ladislav NebeskyÌ

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